A sphere with several labels.

The sphere in the diagram has radius \(r\) and is cut by two parallel planes distance \(h\) apart.

One of these planes is at a distance \(x\) from the “north pole” of the sphere, as shown in the diagram.

What is the surface area of the part of the sphere lying between these two parallel planes?

Perhaps we could try to find some exact or approximate answers in certain special cases to get us started – that might give us some insight into the problem. What values could we pick for \(h\) and/or \(x\) which would make the problem either simpler, or allow us to give a reasonable approximate answer?

You could consider the case \(x=0\). What values of \(h\) would be straightforward in this case?

You could consider the case where \(h\) is very small indeed. What does the part of the sphere between the planes look like in this case? Can you work out the area (approximately) in this case? The next box gives a further suggestion for how to do this.

Does this remind you of the Cones problem?

Once you’ve done this, can you solve the main problem by chopping up the part of the sphere between the planes in a helpful way?